Question

If the focus of a parabola divides a focal chord of the parabola in segments of length 3 and 2, the length of the latus rectum of the parabola is-

• A. $\frac{3}{2}$
• B. $\frac{6}{5}$
• C. $\frac{12}{5}$
• D. $\frac{24}{5}$

Answer 1

The correct option is D

$\frac{24}{5}$

Let $y^2=4ax$ be the equation of the parabola, then vertices of a focal chord of the parabola, then

$t_1{t}_2=–1$. Let SP = 3, SQ = 2

$\text{SP}=\sqrt{a^2(1-t_1^2{)}^2+4a^2{t}_1^2}=a(1+t_1^2)=3$ ...... (i)

and $\text{SQ}=a(1+\frac{1}{t_1^2})=2$ ....... (ii)

From (i) and (ii), we get $t_1^2=\frac{3}{2}$ and $a=\frac{6}{5}$. Hence, the length of the latus rectum $=\frac{24}{5}$.